Chapter 4: Scatterplots and Correlation
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1 Chapter 4: Scatterplots and Correlation Correlation text exr 4.10 pg 108 Ch4-image Ch4 exercises: 4.1, 4.29, 4.39 Most interesting statistical data analysis in the social sciences examines relationships between or among variables: Does drinking beer affect the level of alcohol in our blood? Does advertising increase the demand for a commodity? Are women paid less than men? When we are interested in the relationship between two variables, it is usually because one variable is thought to affect, partly determine, or influence the other. For example, we may expect that individuals education will have an influence on their income; or the demand for a commodity to influence its price. In these cases, we say that income (or demand) is the response variable (also called the dependent variable) and education (or price) is the explanatory variable (also called the independent variable or the predictor). The distinction between explanatory and response variables is clearest in experimental research, such as medical trials or agricultural experiments, where the explanatory variable is directly manipulated by the researcher. If, for example, we could take a group of young children and see to it that some (at random) receive only a grade school education, others a high-school education, and still others a university education, we could more definitively assess the effect of education on income. Sometimes (but much less often) we are interested in the relationship between two variables, for example, verbal and mathematical scores on an achievement test, or average marks in two universities, without identifying one as the explanatory variable and the other as the response. Scatterplots Correlation
2 Chapter 4 Concepts 2 Explanatory and Response Variables Displaying Relationships: Scatterplots Interpreting Scatterplots Adding Categorical Variables to Scatterplots Measuring Linear Association: Correlation Facts About Correlation
3 Scatterplot 4 The most useful graph for displaying the relationship between two quantitative variables is a scatterplot. A scatterplot shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as a point on the graph. How to Make a Scatterplot 1. Decide which variable should go on each axis. If a distinction exists, plot the explanatory variable on the x-axis and the response variable on the y-axis. 2. Label and scale your axes. 3. Plot individual data values. 4
4 Scatterplot 5 Example: Make a scatterplot of the relationship between body weight and pack weight for a group of hikers. Body weight (lb) Backpack weight (lb)
5 Interpreting Scatterplots 6 To interpret a scatterplot, follow the basic strategy of data analysis from Chapters 1 and 2. Look for patterns and important departures from those patterns. How to Examine a Scatterplot As in any graph of data, look for the overall pattern and for striking departures from that pattern. You can describe the overall pattern of a scatterplot by the direction, form, and strength of the relationship. An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship.
6 Interpreting Scatterplots 7 Two variables have a positive association when above-average values of one tend to accompany above-average values of the other, and when below-average values also tend to occur together. Two variables have a negative association when above-average There is one possible outlier, the hiker values of one tend to accompany below-average with values the body of weight the other. of 187 pounds seems to be carrying relatively less weight than are the other group members. Strength Direction Form There is a moderately strong, positive, linear relationship between body weight and pack weight. It appears that lighter hikers are carrying lighter backpacks.
7 Adding Categorical Variables Consider the relationship between mean SAT verbal score and percent of high-school grads taking SAT for each state. To add a categorical variable, use a different plot color or symbol for each category. Southern states highlighted 8
8 Measuring Linear Association 9 A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. The correlation r measures the strength of the linear relationship between two quantitative variables. r is always a number between -1 and 1. r > 0 indicates a positive association. r < 0 indicates a negative association. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 toward -1 or 1. The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.
9 15 Correlation
10 Facts About Correlation Correlation makes no distinction between explanatory and response variables. 2. r has no units and does not change when we change the units of measurement of x, y, or both. 3. Positive r indicates positive association between the variables, and negative r indicates negative association. 4. The correlation r is always a number between -1 and 1. Cautions: Correlation requires that both variables be quantitative. Correlation does not describe curved relationships between variables, no matter how strong the relationship is. Correlation is not resistant. r is strongly affected by a few outlying observations. Correlation is not a complete summary of two-variable data.
This means that the explanatory variable accounts for or predicts changes in the response variable.
Lecture Notes & Examples 3.1 Section 3.1 Scatterplots and Correlation (pp. 143-163) Most statistical studies examine data on more than one variable. We will continue to use tools we have already learned
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